#372910

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

881 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,25)(18,27)(19,32)(21,28)(23,29)(24,31)(26,30)(33,35)(34,36);(1,3,4,2)(5,7,9,20,11,19,10,21)(6,12,15,17,16,28,14,32)(8,23,25,18,26,35,24,34)(13,29,22,36,31,33,30,27);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,22)(15,30)(16,31)(17,33)(19,34)(20,23)(21,35)(28,36)(29,32);(6,14,16,15)(7,19)(12,17,28,32)(13,22,31,30)(18,34)(20,21)(23,35)(27,33,36,29)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280106);